On the degree of semi-stable reduction

TL;DR

This paper investigates bounds on the minimal degree of field extensions needed for an abelian variety over a number field to attain semi-stable reduction, linking it to monodromy groups and classical bounds.

Contribution

It provides a new explicit bound on the minimal degree for semi-stable reduction based on monodromy group sizes and demonstrates the bound's near-optimality through local coverings.

Findings

Bound on $d(A)$ in terms of monodromy group cardinalities

Explicit relation to Minkowski bound

Bound is tight up to its 2-part

Abstract

For an abelian variety $A$ over a number field we study bounds depending only on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of the cardinalities of the finite monodromy groups of $A$ which leads to a bound on $d(A)$ in terms of the classical Minkowski bound. We then show this bound is tight up to its $2$-part by considering $p$-adic coverings of the local points of a universal abelian scheme.